Non-Gaussianity of superhorizon curvature perturbations beyond $\delta$ N formalism

Abstract

We develop a theory of nonlinear cosmological perturbations on superhorizon scales for a single scalar field with a general kinetic term and a general form of the potential. We employ the ADM formalism and the spatial gradient expansion approach, characterised by $O({\epsilon }^m)$, where $\epsilon =1/(HL)$ is a small parameter representing the ratio of the Hubble radius to the characteristic length scale $L$ of perturbations. We obtain the general solution for a full nonlinear version of the curvature perturbation valid up through second-order in $\epsilon$ ($m=2$). We find the solution satisfies a nonlinear second-order differential equation as an extension of the equation for the linear curvature perturbation on the comoving hypersurface. Then we formulate a general method to match a perturbative solution accurate to $n$-th-order in perturbation inside the horizon to our nonlinear solution accurate to second-order ($m=2$) in the gradient expansion on scales slightly greater than the Hubble radius. The formalism developed in this paper allows us to calculate the superhorizon evolution of a primordial non-Gaussianity beyond the so-called $\delta N$ formalism or separate universe approach which is equivalent to leading order ($m=0$) in the gradient expansion. In particular, it can deal with the case when there is a temporary violation of slow-roll conditions. As an application of our formalism, we consider Starobinsky's model, which is a single field model having a temporary non-slow-roll stage due to a sharp change in the potential slope. We find that a large non-Gaussianity can be generated even on superhorizon scales due to this temporary suspension of slow-roll inflation.

27 pages, 3 figures

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• Review on stochastic approach to inflation
• Stochastic inflation at all order in slow-roll parameters: foundations
• Consistency conditions and primordial black holes in single field inflation
• Conservation of the nonlinear curvature perturbation in generic single-field inflation
• CMB $\mu T$ cross-correlations as a probe of PBH scenarios
• Anisotropic separate universe and Weinberg's adiabatic mode
• Full nonlinear growing and decaying modes of superhorizon curvature perturbations
• On the Hamilton-Jacobi approach to inflation beyond slow roll
• Manifestly gauge invariant theory of the nonlinear cosmological perturbations in the leading order of the gradient expansion
• Dip and non-linearity in the curvature perturbation from inflation with a transient non-slow-roll stage